Question:medium

What is the value of \(\log\log\sqrt{\sqrt[5]{\sqrt[10]{\sqrt[10]{10}}}}\)?

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Multiply all the root indices together: \(2\times5\times10\times10=1000\), so the whole radical equals \(10^{1/1000}\); then take log twice.
Updated On: Jul 20, 2026
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The Correct Option is A

Solution and Explanation

Instead of peeling one root at a time, combine all four root indices in one shot.

A chain of nested roots $\sqrt[n_1]{\sqrt[n_2]{\sqrt[n_3]{\sqrt[n_4]{x}}}}$ always equals $x^{1/(n_1 n_2 n_3 n_4)}$, because each root just multiplies the fractional exponent.

Here the chain (from outside in) is square root (index 2), fifth root (index 5), tenth root (index 10), tenth root (index 10), applied to 10. So the product of indices is:
$2 \times 5 \times 10 \times 10 = 1000$

Therefore the whole nested radical simplifies directly to:
$10^{1/1000}$

Now apply log base 10 once:
$\log\left(10^{1/1000}\right) = \dfrac{1}{1000} = 10^{-3}$

And apply log base 10 a second time:
$\log\left(10^{-3}\right) = -3$

\[\boxed{-3}\]
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